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Deterministic Distributed Edge-Coloring Via Hypergraph Maximal Matching

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Deterministic Distributed Edge-Coloring Via Hypergraph Maximal Matching Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching Manuela Fischer Mohsen Ghaffari Fabian Kuhn ETH Zurich ETH Zurich University of Freiburg manuela.fi[email protected] ghaff[email protected] [email protected] Abstract We present a deterministic distributed algorithm that computes a (2∆ 1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree ∆, in O(log− 7 ∆ log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms· from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities O(√log n) were 2 by Panconesi and Srinivasan [STOC’92] and O˜(√∆) + O(log∗ n) by Fraigniaud, Heinrich, and Kosowski [FOCS’16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2∆ 1)-edge-coloring to poly(log log n) rounds. − The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r — where each hyperedge has at most r vertices — with n nodes and maximum degree ∆, this algorithm computes a maximal matching in O(r5 log6+log r ∆ log n) rounds. · This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin’s book, a (log ∆/ε)O(log 1/ε) -round deterministic algorithm for (1 + ε)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting λ-arboricity graphs with out-degree at most (1 + ε)λ , for any constant ε> 0, hence partially answering Open Problem 10 of Barenboim and Elkin’s⌈ book.⌉ arXiv:1704.02767v1 [cs.DS] 10 Apr 2017 1 Introduction and Related Work Distributed graph algorithms have been studied extensively over the past 30 years, since the seminal work of Linial [Lin87]. Despite this, determining whether there are efficient deterministic distributed algorithms for the most classic problems of the area remains a long-standing open question. Distributed graph algorithms are typically studied in a standard synchronous message passing model known as the LOCAL model [Lin87, Pel00]: the network is abstracted as an undirected graph G = (V, E), n = V , with maximum degree ∆, and where each node has a Θ(log n)-bit unique identifier. Initially,| nodes| only know their neighbors in G. At the end, each node should know its own part of the solution, e.g., the colors of its edges in edge-coloring. Communication happens in synchronous rounds, where in each round each node sends a message to each of its neighbors.1 The main complexity measure is the number of rounds needed for solving a given graph problem. The four classic local distributed graph problems are maximal independent set (MIS), (∆ + 1)- vertex-coloring, (2∆ 1)-edge-coloring, and maximal matching [PR01, BE13]. All of these problems − have trivial greedy sequential algorithms, as well as simple O(log n)-round randomized distributed algorithms [Lub86,ABI86], and even some faster ones [BEPS12,EPS15,Gha16,HSS16]. But the deter- ministic distributed complexity of these problems remains widely open, despite extensive interest (see, e.g., the first five open problems of [BE13]). Particularly, with regards to MIS — which is the hardest of the four problems, as the other three can be reduced to MIS locally [Lin87] — Linial [Lin92] asked “can it [MIS] always be found [deterministically] in polylogarithmic time?”. This remains the most well-known open question of the area. The best known round complexity is 2O(√log n), due to Panconesi and Srinivasan [PS92]. Panconesi and Rizzi pointed out in the year 2000 [PR01] that “while maximal matchings can be computed in polylogarithmic time, in n, in the distributed model [HKP98], it is a decade old open problem whether the same running time is achievable for the remaining 3 structures.” The status remains the same as of today, after almost two more decades. In particular, for edge-coloring which is our main target, Barenboim and Elkin stated the following problem in their recent Distributed Graph Coloring book [BE13]: Open Problem 11.4 [BE13] Devise or rule out a deterministic (2∆ 1)-edge-coloring algorithm that runs in polylogarithmic time. − 1.1 Our Contributions Improved Deterministic Edge-Coloring Algorithm: One of our main end results is a positive resolution of the above question. Theorem 1.1. There is a deterministic distributed algorithm that computes a (2∆ 1)-edge-coloring in − O(log7 ∆ log n) rounds, in any n-node graph with maximum degree ∆. Moreover, the same algorithm solves list-edge-coloring,· where each edge e E must get a color from an arbitrary given list L of ∈ e colors with L = d + 1, where d denotes the number of edges incident to e. | e| e e For list-edge-coloring, the previously best known round complexity was 2O(√log n), by a classic net- O(√log n log log n) work decomposition of Panconesi and Srinivasan [PS92], which itself improved on an 2 · - round algorithm of Awerbuch et al. [ALGP89]. For low-degree graphs, the best known is an (O˜(√∆)+ O(log∗ n))-round algorithm of Fraigniaud, Heinrich, and Kosowski [FHK16], which is more general and applies also to (∆ + 1)-list-vertex-coloring. There are some known poly log n-round determin- istic edge-coloring algorithms, but all have two major shortcomings: (A) they require more colors, and (B) they are quite restricted and do not work for list-coloring. These algorithms are as follows: 1In the LOCAL model, messages might be of arbitrary size. A variant of the model where the messages have to be of bounded size is known as the CONGEST model [Pel00]. Our edge-coloring algorithms and our MIS and vertex-coloring algorithms for graphs of bounded neighborhood independence in fact work with small O(log n)-bit size messages. Though, for the sake of the readability, we avoid explicitly discussing the details of this aspect. 1 O log ∆ (I) a ((2 + o(1))∆)-edge-coloring by Ghaffari and Su [GS17]; (II) a ∆ 2 log log ∆ -edge-coloring by · Barenboim and Elkin [BE11]—see also [GS17, Appendix B] for a short proof; and (III) an O(∆ log n)- edge-coloring by Czygrinow et al. [CHK01]. See also [BE11, Chapter 8]. We also note that a recent work of Barenboim, Elkin, and Maimon [BEM16] presents an efficient deterministic algorithm for computing a (∆ + o(∆))-edge-coloring in graphs with arboricity a ∆1 δ, ≤ − for some constant δ > 0. In Corollary 5.1, we sketch how a simple combination of the list-edge-coloring algorithm of Theorem 1.1 with H-partitionings [BE13, Chapter 5.1] significantly extends their result. Improved Randomized Edge-Coloring Algorithm: The deterministic list-edge-coloring algo- rithm of Theorem 1.1, in combination with some randomized edge-coloring algorithms of [EPS15, BEPS12], also improves the complexity of randomized algorithms for (2∆ 1)-edge-coloring, making it the first among the four classic problems whose randomized complexity falls− down to poly(log log n). Corollary 1.2. There is a randomized distributed algorithm that computes a (2∆ 1)-edge-coloring in O(log8 log n) rounds, with high probability, in any n-node graph with maximum degree− ∆. The previous (worst-case) complexity for randomized (2∆ 1)-edge-coloring was 2O(√log log n) − rounds2, due to Elkin, Pettie, and Su [EPS15]. By improving this, Corollary 1.2 widens the prov- able gap between the complexity of (2∆ 1)-edge-coloring, which is now in poly(log log n) rounds, and the complexity of maximal matching,− which needs Ω( log n/ log log n) rounds [KMW16]. Unified Formulation as Hypergraph Maximal Matchingp : Our first step towards proving Theorem 1.1 is a simple unification of all the aforementioned four classic problems: MIS, (∆ + 1)- vertex-coloring, (2∆ 1)-edge-coloring, and maximal matching. We can cast each of these problems − as a maximal matching problem on hypergraphs3 of some rank r, which depends on the problem, and increases as we move from maximal matching to maximal independent set. In other words, the hy- pergraph maximal matching problem can be used to obtain a smooth interpolation between maximal matching in graphs and maximal independent set in graphs. Recall that the rank of a hypergraph is the maximum number of vertices in any of its hyperedges. Moreover, a matching in a hypergraph is a set of hyperedges, no two of which share an endpoint. We present a reduction from (2∆ 1)-edge-coloring to maximal matching in rank-3 hypergraphs, − as we sketch next in Lemma 1.3. A similar reduction can be used for list-edge-coloring, as formalized in Lemma 2.13. We note that these reductions are inspired by the well-known reduction of Luby from (∆ + 1)-vertex-coloring to MIS [Lub86, Lin87]. Lemma 1.3. Given a deterministic distributed algorithm that computes a maximal matching in A N-vertex hypergraphs of rank 3 and maximum degree d in T (N, d) rounds, there is a deterministic distributed algorithm that computes a (2∆ 1)-edge-coloring of any n-node graph G = (V, E) B − with maximum degree ∆ in at most T (3n∆, 2∆ 1) rounds. − Proof Sketch. To edge-color G = (V, E), we generate a hypergraph H: Take 2∆ 1 copies of G. − For each edge e E, let e1 to e2∆ 1 be its copies.
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